To find the adjoint of a matrix, M, the following transformation is applied: take the transpose of the matrix and then take the complex conjugate of all elements of the matrix. The resulting matrix is called the adjoint of M and is denoted by $M^{\mathit{\*}}$.

Note that if all entries of M are real numbers then $M^t\mathit{\=}\mathit{ }{M}^{\mathit{\*}}$ because each entry is the complex conjugate of itself.

We will compute both $M \cdot M^{\mathit{\*}}$and ${\mathrm{M}}^{\*}\cdot M$ and check if they are equal or not.

M$$M^{\mathit{\*}}$=  and $M^{\mathit{\*}}\mathit{\cdot }M$ = .
The matrix normal.

We will calculate the eigenvalues of the matrix by finding the matrix's characteristic polynomial. We solve:  
The characteristic polynomial for the matrix is:  This gives eigenvalues  with multiplicities of , where the left side of each equation is the eigenvalue and the right side of each equation is the multiplicity of that eigenvalue.

The next step is to find the eigenvectors for the matrix M. This can be done manually by finding the solutions for v in the equation $\left(M - \mathrm{\λ}\cdot I \right)\cdot v \= 0$ for each of the eigenvalues $\mathrm{\lambda }$ of M. To solve this manually, the equation will give a system of equations with the number of variables equal to the number of the dimensions of the matrix.

Choose the eigenvalue for which you want to find eigenvectors. $\mathrm{\λ}$ = Choose4-2 

An aid for solving $\left(M - \mathrm{\lambda }\cdot I \right)\cdot v \= 0$ is augmenting M into a matrix A that has an extra column of zeros in the end. Then, applying Gaussian Elimination on the augmented matrix A and transforming it into Reduced Row Echelon Form to simplify the calculations.

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Then, transform the matrix into a system of linear equations and solve it. Note that $\left\{\mathrm{x\_\_1}\, \mathrm{x\_\_2}\, ..\. \, \mathrm{x\_\_n}\right\}$ represent the variables for the system in the order: $\left[\begin{array}{c}\mathrm{x\_\_1}\\ \mathrm{x\_\_2}\\ ⋮\\ x_n\end{array}\right]$.

System:  Solution (In vector form):  . Therefore, an orthonormal basis for the eigenspace $\mathrm{E\_\_\λ}$ is (by using Gram-Schmidt): .
Therefore, combining the results for all eigenvalues we get that an orthonormal basis for ${\mathrm{\ℝ}}^n$ is:.

The diagonal matrix to which M is similar to is the diagonal matrix that has M's eigenvalues as entries. Note that, the eigenvalues are repeated in terms of multiplicity and equal eigenvalues need to be adjacent to each other.

So we get

D =

The transition matrix will be the concatenation of the eigenvectors beside each other depending on the order of the eigenvalues.

Q  =